What vitakka actually means?Initial thought, Application of thought, mental factor...Different label of words gave different impression towards its meaning, looking forward to understand more about it.

Secondary inquiries:What gave rise to vitakka?Can vitakka arise without the mind coming into contact with external objects or mental objects?

No contact with external objects or mental objects would be what, Neither perception nor non-perception? Traditionally vitakka is said to cease with 2nd jhana, you don't need to keep directing though to the breath or other object.

... Herein, applied thinking (vitakkama) is applied thought (vitakka); hitting upon, is what is meant. It has the characteristic of directing the mind onto an object (mounting the mind on its object). Its function is to strike at and thresh- for the mediator (2 The Visuddhimagga deals with vitakka in the section on samatha. The meditator is someone who cultivates samatha.) is said, in virtue of it, to have the object touched and struck at by applied thought. It is manifested as the leading of the mind onto an object...The Atthasalini (Book I, Part IV, Chapter I, 114) gives a similar definition. This commentary uses a simile of someone who wants to "ascend" the king's palace and depends on a relative or friend dear to the king to achieve this. In the same way the citta which is accompanied by vitakka depends on the latter in order to "ascend" to the object, to be directed to the object. Vitakka leads the citta to the object so that citta can cognize it.

What gave rise to vitakka?Can vitakka arise without the mind coming into contact with external objects or mental objects?

Vitaka arises with most cittas, except for the 5 pairs of sense consciousness (seeing, hearing, smelling, tasting, touching- which can be the result of either wholesome or unwholesome kamma), and the jhana cittas from the second stage up.

Vitaka arises because of numerous conditions: kamma condition; vipaka condition ; conascence condition etc....It doesn't arise at someone's will. In case of the development of samatha, when it hist again and again the same object, it arises because of natural decisive condition (in particular object-natural decisive condition), among other conditions. Understanding more details about this allow us to understand more about anattaness and about the true nature of samatha bhavana....

Vitaka has to arise with citta, since cetasikas can only arise together with citta. When a citta arises, it arises to experience an object, be it nama, rupa or just a concept, or Nibanna. So vitaka never arises without citta, and it always experiences an object.

From what i red, vitakka is abandoned when one advanced into the 2nd Jhana and forwards, yet in the Gavi sutta, it sounded like thoughts still arise along the higher practices.

"The thought occurs to him, 'What if, with the stilling of directed thoughts & evaluations, I were to enter & remain in the second jhana: rapture & pleasure born of composure, unification of awareness free from directed thought & evaluation — internal assurance.' Without jumping at the second jhana, he — with the stilling of directed thoughts & evaluations — enters & remains in the second jhana: rapture & pleasure born of composure, unification of awareness free from directed thought & evaluation — internal assurance. He sticks with that theme, develops it, pursues it, & establishes himself firmly in it.

"The thought occurs to him, 'What if, with the fading of rapture, I... were to enter & remain in the third jhana...' Without jumping at the third jhana, with the fading of rapture, he remains equanimous, mindful, & alert, and senses pleasure with the body. He enters & remains in the third jhana, of which the Noble Ones declare, 'Equanimous & mindful, he has a pleasant abiding.' He sticks with that theme, develops it, pursues it, & establishes himself firmly in it.

"The thought occurs to him, 'What if I... were to enter & remain in the fourth jhana...' Without jumping at the fourth jhana, with the abandoning of pleasure & stress — as with the earlier disappearance of elation & distress — he enters & remains in the fourth jhana: purity of equanimity & mindfulness, neither-pleasure-nor-pain. He sticks with that theme, develops it, pursues it, & establishes himself firmly in it. http://www.accesstoinsight.org/tipitaka ... .than.html

If the mind in higher Jhanas is said to be without vitakka and conceptual thoughts, then how can the mind sticks with the theme( which must have arisen from thoughts) ?And without vitakka and thoughts, what are the factors for the mind to exit Jhana?

If you have asked me of the origination of unease, then I shall explain it to you in accordance with my understanding: Whatever various forms of unease there are in the world, They originate founded in encumbering accumulation. (Pārāyanavagga)

Exalted in mind, just open and clearly aware, the recluse trained in the ways of the sages:One who is such, calmed and ever mindful, He has no sorrows! -- Udana IV, 7